Non-additive ((6,2,3)) qubit code with transversal T

Problem

Does there exist a genuinely non-additive two-dimensional six-qubit code of distance three on which a tensor product of single-qubit unitaries implements the logical \(T\) gate exactly? Write \(\mathcal P_n:=\{I,X,Y,Z\}^{\otimes n}\) for the phase-free Hermitian Pauli basis, where the weight \(\operatorname{wt}(E)\) of \(E\in\mathcal P_n\) counts its nonidentity tensor factors. An \(((n,K,d))_2\) code is a \(K\)-dimensional subspace of \((\mathbb C^2)^{\otimes n}\) that detects every Pauli operator of weight less than \(d\); degenerate codes are allowed. A \(((6,2,3))_2\) code is thus the range of an isometry \(V:\mathbb C^2\to(\mathbb C^2)^{\otimes6}\), with code projector \(P:=VV^\dagger\), satisfying

\begin{equation} V^\dagger V=I_2, \qquad V^\dagger EV=c_EI_2 \quad\text{for all }E\in\mathcal P_6\text{ with }\operatorname{wt}(E)<3, \qquad c_E\in\mathbb R. \tag{1} \end{equation}

The code is genuinely non-additive if no tensor product of single-qubit unitaries, even combined with a permutation of the qubits, maps it onto a Pauli stabilizer code, that is, onto the joint \(+1\) eigenspace of an abelian subgroup of the Pauli group. The logical gate and its transversal realization are

\begin{equation} T:=\operatorname{diag}(1,e^{i\pi/4}), \qquad \Bigl(\bigotimes_{j=1}^{6}U_j\Bigr)V=e^{i\phi}VT, \qquad U_1,\ldots,U_6\in U(2),\quad \phi\in\mathbb R. \tag{2} \end{equation}

The local factors \(U_j\) may differ from one another and need not be physical \(T\) gates. No ancillary qubits, measurements, code switching, or additional transversal logical gates are assumed. The question asks whether some isometry satisfying Eq. (1) has a genuinely non-additive range and admits unitaries \(U_1,\ldots,U_6\) and a phase \(\phi\) satisfying Eq. (2).

Source

Derived from a documented gap. Zhang, Wu, Huang, and Zeng realize transversal \(T\) on seven qubits (Section 6.3.1), and their Table 2 lists the cyclic group \(\mathrm C_{16}\), generated by the determinant-one representative of \(T\), among the transversal groups for which no \(((6,2,3))\) code was found [ZWHZ25]. The source neither poses the six-qubit question explicitly nor claims nonexistence; the requirement of genuine non-additivity belongs to the present formulation.

Progress

By the Knill–Laflamme conditions, Eq. (1) is equivalent to exact correction of the error set \(\mathcal F:=\{I^{\otimes6}\}\cup\{E\in\mathcal P_6:\operatorname{wt}(E)=1\}\),

\begin{equation} PF_a^\dagger F_bP=\Lambda_{ab}P \qquad(F_a,F_b\in\mathcal F), \tag{3} \end{equation}

because the products \(F_a^\dagger F_b\) span the operators of weight at most two. The question therefore concerns arbitrary single-qubit errors, not a particular noise channel [KL97].
Du, Zhang, Poon, and Zeng introduced the signature norm of a code. For the rank-two projector \(P_n\) of an \(((n,2,3))_2\) code its square is

\begin{equation} s(P_n)^2:=\frac14 \sum_{\substack{E\in\mathcal P_n\\ 1\leq\operatorname{wt}(E)\leq2}} \lvert\operatorname{Tr}(P_nE)\rvert^2 . \tag{4} \end{equation}

They prove that it is invariant under local unitaries; it is also invariant under qubit permutations. For a stabilizer code each coefficient \(\operatorname{Tr}(P_nE)/2\) lies in \(\{0,\pm1\}\), so \(s(P_n)^2\) is an integer (Appendix C of the arXiv version). A noninteger value of Eq. (4) therefore certifies genuine non-additivity in the sense of the statement [DZPZ26].
Anglès Munné and Huber give an exact rational semidefinite-programming certificate that every \(((6,2,3))_2\) code is impure (Section 4.2, item (e); Table 1 marks the optimal size \(K=2\) at block length \(6\) and distance \(3\) as impure). Equivalently, \(s(P_6)^2>0\) for every isometry satisfying Eq. (1), so some Pauli operator of weight one or two has a nonzero expectation value on any candidate code [AMH26].
Seven qubits suffice. Section 6.3.1 of Zhang, Wu, Huang, and Zeng gives an explicit exact \(((7,2,3))_2\) code with two free phases, \(\lvert1_L\rangle=X^{\otimes7}\lvert0_L\rangle\), and angle vector \((1,2,2,2,2,3,3)\): the operator \(X^{\otimes7}\) and a tensor product of single-qubit \(Z\) rotations implement logical \(X\) and, up to global phase, \(T^\dagger=\operatorname{diag}(1,e^{-i\pi/4})\). Hence \(T=(T^\dagger)^7\) is transversal as well. The example has

\begin{equation} s(P_7)^2=\frac{21}{8}, \tag{5} \end{equation}

so Eq. (5) certifies that it is genuinely non-additive [ZWHZ25], [DZPZ26].
Six qubits already support a different non-Clifford phase. Section 4.1 of the same paper gives an exact \(((6,2,3))_2\) code with transversal logical \(\operatorname{diag}(1,e^{-2\pi i/5})\), up to global phase, and signature norm \(\sqrt{0.84}\), that is, \(s(P_6)^2=21/25\); this code is therefore genuinely non-additive. Its logical phase has projective order \(5\), whereas \(T\) has projective order \(8\). Table 2 lists \(\mathrm C_{16}\), generated by \(e^{-i\pi/8}T\), among the targets for which no \(((6,2,3))\) code was found; the caption states only that no such code has been found so far [ZWHZ25].
He, Lu, and Zeng (arXiv version 3, April 2026) exactly resolve the twelve seven-qubit candidates that survive the subset-sum and linear-programming filters in the complementary binary-dihedral ansatz, in which the physical gates are diagonal phase gates with angles in multiples of \(2\pi/8\) and \(\lvert1_L\rangle=X^{\otimes7}\lvert0_L\rangle\):

\begin{equation} 12\ \text{filtered candidates} =10\ \text{realizable candidates}+2\ \text{excluded candidates}. \tag{6} \end{equation}

Section IV and Appendices F–G give the exact constructions and no-go proofs, which the authors report as verified in Lean. This restricted seven-qubit classification does not settle the unrestricted six-qubit question; the same paper’s catalogue on at most six qubits concerns distance two [HLZ25].

Comment

Literature checked through 15 September 2026: no exact six-qubit code satisfying Eqs. (1) and (2) and no general exclusion was located. This is a finite-size feasibility problem derived from the unfilled \(\mathrm C_{16}\) entry of Table 2 in [ZWHZ25], not a named nonexistence conjecture. The seven-qubit constructions, the restricted seven-qubit classification, the distance-two catalogues, and the numerical six-qubit optimizations do not decide it. A negative answer must cover arbitrary local factors \(U_j\) and every \(((6,2,3))_2\) code subspace, each of which is necessarily impure; a positive answer must also certify genuine non-additivity, for example through a noninteger value of Eq. (4).

References

[KL97]: E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Physical Review A 55, 900–911 (1997).DOI arXiv
[DZPZ26]: M. Du, C. Zhang, Y. T. Poon, and B. Zeng, “Characterizing quantum codes via the coefficients in Knill–Laflamme conditions,” npj Quantum Information 12, 13 (2026).DOI arXiv
[ZWHZ25]: C. Zhang, Z. Wu, S. Huang, and B. Zeng, “Transversal Gates in Nonadditive Quantum Codes,” arXiv:2504.20847 (2025).DOI arXiv
[AMH26]: G. Anglès Munné and F. Huber, “SDP bounds on quantum codes: rational certificates,” arXiv:2603.19901 (2026).DOI arXiv
[HLZ25]: X. He, S. Lu, and B. Zeng, “Co-Designing Quantum Codes with Transversal Diagonal Gates via Multi-Agent Systems,” arXiv:2510.20728 (2025), version 3 (2026).DOI arXiv

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